Question

For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. $$0.3+0.03+0.003+\cdots$$

Short answer

Answer: The value of the infinite series \(0.3 + 0.03 + 0.003 + \cdots\) is \(\frac{1}{3}\).

Step-by-step solution

Identify the Infinite Series Formula

We can represent the given infinite series as a geometric series since each term is a constant multiple of the previous one. The general formula for a geometric series is given by: $$S_n = a\frac{1-r^n}{1-r}$$ where \(S_n\) denotes the sum of the first n terms, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms considered. For our given series, \(a = 0.3\) and \(r = 0.1\). We will use this formula to find the first four partial sums for the sequence.

Calculate the First Four Partial Sums

We will now find the first four partial sums (\(S_1\), \(S_2\), \(S_3\), and \(S_4\)) using the formula derived in Step 1: 1. \(S_1 = 0.3\frac{1-0.1^1}{1-0.1} = 0.3\) 2. \(S_2 = 0.3\frac{1-0.1^2}{1-0.1} = 0.3 + 0.03 = 0.33\) 3. \(S_3 = 0.3\frac{1-0.1^3}{1-0.1} = 0.3 + 0.03 + 0.003 = 0.333\) 4. \(S_4 = 0.3\frac{1-0.1^4}{1-0.1} = 0.3 + 0.03 + 0.003 + 0.0003 = 0.3333\)

Analyze the Pattern

Upon analyzing the first four partial sums we calculated above, we observe that the sequence of partial sums approaches \(0.3333\cdots\) as we add more terms. This suggests that the value of the infinite series will be equal to this value.

Make a Conjecture about the Infinite Series

Based on our analysis of the first four partial sums, we can make a conjecture that the value of the given infinite series is: $$0.3 + 0.03 + 0.003 + \cdots = 0.3333\cdots$$ This is a repeating decimal and can be written as a fraction, in this case, \(\frac{1}{3}\).

Key concepts

Geometric Series

A geometric series is a type of infinite series that consists of a sequence of terms where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In mathematical terms, a geometric series can be expressed as:

\begin{align*} a + ar + ar^2 + ar^3 + \text{\textellipsis}end{align*}where \( a \) is the first term and \( r \) is the common ratio. If the common ratio is between -1 and 1 (excluding 0), the series has a sum to which it converges. This type of series is quite prevalent in mathematical finance, physics, and computer science due to its simple structure and the ease of computing its sum using a basic formula.

For example, the geometric series in the provided exercise is \( 0.3 + 0.03 + 0.003 + \text{\textellipsis} \), with a common ratio of 0.1. Understanding this concept is crucial for students as it helps to solve many problems involving repetitive or proportional growth or decay.

Partial Sums

The notion of partial sums is essential when dealing with infinite series. A partial sum is the sum of the first \( n \) terms of a series. It is a technique used to understand better and visualize the behavior of the total sum as more terms are added. For an infinite series, we look at the series of partial sums to determine whether the infinite series converges to a specific value or not.

The partial sums are denoted by \( S_1, S_2, S_3, \text{\textellipsis}, S_n \), where each subscript signifies how many terms of the series are being summed. In our textbook example, the first four partial sums of \( 0.3 + 0.03 + 0.003 + \text{\textellipsis} \) are calculated and shown to be approaching a certain value, indicating the series' behavior as more terms are included. Contrasting the sequence of partial sums often leads to insights regarding convergence or divergence of the series.

Convergence of Series

The convergence of a series is a fundamental topic in calculus that deals with determining whether the sum of an infinite series approaches a finite value. When the series converges, the sequence of partial sums tends closer to a specific number as more terms are added. Conversely, if a series does not approach any limit and indefinitely increases or oscillates, then the series diverges.

To establish convergence, various tests can be applied, such as the ratio test, the root test, or for geometric series, simply examining the common ratio. For a geometric series like that in our exercise, if the absolute value of the common ratio \( r \) is less than 1, the series converges. In our example, the absolute value of the common ratio, 0.1, is less than 1, so we can conclude that the series converges. The calculated partial sums suggest that the series converges to \( \frac{1}{3} \). This understanding is crucial because it allows mathematicians and scientists to make sense of infinite processes, ensuring that calculations involving these series are meaningful and can be applied in various practical situations.